Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables

TL;DR

This paper explores the structure of symmetric polynomials in non-commuting variables, revealing a tensor product decomposition analogous to classical reflection group theorems, and relates it to its commutative counterpart.

Contribution

It introduces a tensor product decomposition of the algebra of non-commutative symmetric polynomials, extending classical invariant theory results to non-commuting variables.

Findings

Realized the commutative symmetric polynomials as invariants within the non-commutative algebra.

Discovered a tensor product decomposition similar to Chevalley and Shephard-Todd theorems.

Established a connection between the non-commutative and commutative symmetric polynomial algebras.

Abstract

We analyze the structure of the algebra N of symmetric polynomials in non-commuting variables in so far as it relates to its commutative counterpart. Using the "place-action" of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of N analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups.